260 ANALYTICAL MECHANICS When e = Q and tan a. < AC, tan /3 = — oo and /3 = — - ; therefore the particle 2i is reflected towards the left and slides along the plane. PROBLEMS. 1. A perfectly elastic ball impinges obliquely on another ball at rest. Prove that their masses are equal if, after impact, the balls move at right angles. 2. A billiard ball strikes simultaneously two billiard balls at rest, and comes to rest. Show that the coefficient of restitution is f . 3. A particle slides down a smooth inclined plane and then rebounds from a horizontal plane. Find the range of the first rebound. 4. A bullet strikes a target at 45° and rebounds at the same angle. Prove that e = — — -, where ju is the coefficient of friction. 6. Four smooth rods, which form a square, are fixed on a smooth horizontal plane. A particle which is projected from one corner of the square strikes an adjacent corner after three reflections; show that e (I + e) tan a •• 1 + e (1 + e) where a is the angle the initial velocity makes with the rod joining the two corners and e is the coefficient of restitution. 6. In the preceding problem discuss the values of a for special values of e. 7. Derive an expression for the percentage of energy lost during oblique impact (a) when the contact is smooth; (b) when the contact is rough. 8. Two billiard balls which are in contact are struck, simultaneously, by a third ball moving with a velocity v, in a direction perpendicular to the line of centers of the first two. Supposing the table to be perfectly smooth find the velocity of each ball after impact. 9. In the preceding problem obtain the expression for the loss of energy and find its value for the following special cases. The balls weigh 6 ounces each. (a) v = 16 feet per second, e = 0.8. (b) v — 20 feet per second, e = 0.5. 10. A ball impinges against another ball which has twice as large a mass and is at rest. The smaller ball has a velocity of 60 feet per second in a direction which makes 135° with the line of centers. Find the velocities after impact; e — 0.5.